Tile Shift: read the starting distance without mistaking it for the answer
A four-by-four example explains the difference between a distance floor, a proved shortest route and an impossible arrangement the generator will never deal.
Worked examples use the Standard board unless a different size is stated. Hard and Expert may change the board dimensions or amount of information to track.
Read the word starting literally
Hard and Expert show a starting distance lower bound beside the move counter. It describes the board before you move. It is not a live estimate that decreases after every slide, and it is not a promise that the puzzle can be finished in exactly that many moves. Your current move count answers a different question: how many accepted slides are in your current route, after accounting for Undo.
The distance adds the horizontal and vertical separation of each numbered tile from its goal. The empty square is excluded. A legal slide moves one numbered tile one square, so it can reduce that total by at most one. A starting total of 24 therefore rules out finishing in 23 slides, although it does not establish a 24-slide solution.
Calculate a small four-by-four example
Use slashes to separate rows and consider 1 2 3 4 / 5 6 7 8 / 9 10 blank 12 / 13 14 11 15. Only tiles 11 and 15 are displaced. Tile 11 sits one row below its target; tile 15 sits one column to its right. Their distances add to two, while every other numbered tile contributes zero.
Move the blank Down, bringing 11 upward, then Right, bringing 15 left. The board is solved in two slides. Here the lower bound and an actual two-slide route agree, proving that route shortest. This is a separate teaching position; Hard and Expert deliberately begin farther away, with starting floors of 24 and 36 respectively.
Understand what the sum leaves out
The calculation treats tiles independently. It does not charge for arranging the blank on the useful side of a tile, or for temporarily moving a correct tile to make a corridor. Those preparations can add slides beyond the lower bound. A move that raises the distance can therefore belong to a productive route rather than being an automatic mistake.
Use the number to understand the scale of the initial problem. For your next decision, identify a tile, its intended destination and the square the blank must reach. Rejecting every move that looks farther from the goal can prevent necessary preparation. The game offers Undo for comparing routes, but does not display a complete shortest-route solution.
Distance alone does not prove reachability
Imagine the finished four-by-four board with only tiles 14 and 15 exchanged. Their combined distance is two, yet no legal sequence restores that arrangement. An inversion is a pair of numbered tiles appearing in reverse numerical order when read row by row, ignoring the blank. The exchanged tiles create one inversion; the blank is one row from the bottom. On a reachable four-by-four board, the inversion count plus that bottom-based blank row is odd. This exchanged position gives an even sum.
You do not need to calculate that invariant during ordinary play. New puzzle produces boards through legal slides, so its arrangements remain reachable. The example explains why a small distance is neither a universal solvability test nor a complete difficulty rating. Read the starting floor as a genuine constraint, then build a route whose individual preparations you can explain.
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